Weighted finite difference methods for a nonlinear Klein-Gordon equation with high oscillations in space and time
This work addresses the numerical challenge of solving highly oscillatory wave equations for researchers in computational physics and applied mathematics, providing methods that avoid ε-dependent time step restrictions.
The paper develops explicit and implicit weighted finite difference methods for a nonlinear Klein-Gordon equation with high oscillations in space and time, achieving second-order accuracy and uniform convergence for the scaling parameter ε from arbitrarily small to moderately bounded values.
We consider a nonlinear Klein-Gordon equation in the nonrelativistic limit regime with initial data in the form of a modulated highly oscillatory exponential. In this regime of a small scaling parameter $\varepsilon\ll 1$, the solution exhibits rapid oscillations in both time and space. The solution is approximated, up to $\mathcal{O}(\varepsilon)$, by a superposition of two polarized solutions, which are wave packets that move with opposite group velocities proportional to $\varepsilon^{-1}$. The equations for polarized solutions are formulated in co-moving coordinates and are then discretized by an explicit and an implicit exponentially weighted finite difference method. While the explicit weighted leapfrog method needs to satisfy a CFL-type stability condition, the implicit weighted Crank-Nicolson method is unconditionally stable. Both methods achieve second-order accuracy with time steps and mesh sizes that are not restricted in magnitude by $\varepsilon$. For the approximation of polarized solutions, the methods are uniformly convergent in the range from arbitrarily small to moderately bounded $\varepsilon$. Numerical experiments illustrate the theoretical results.